Question

Let \(X\) follow \(N(3,1)\). Then the value of \(E\big(X^4(X-3)\big)\) equals ______ (answer in integer).

Hint:

Write \(Z=X-3\sim N(0,1)\) and expand \((Z+3)^4Z\); only the even-power terms in \(Z\) survive after taking expectation.

Answer 1

Correct Answer: 144

Step 1: Standardize.
\(Z=X-3\sim N(0,1)\), \(X=Z+3\).

Step 2: Rewrite target.
\[ E\big(X^4(X-3)\big)=E\big((Z+3)^4\cdot Z\big) \]

Step 3: Expand.
\[ (Z+3)^4=Z^4+12Z^3+54Z^2+108Z+81 \]

Step 4: Multiply by Z, take expectation.
\[ E\big((Z+3)^4 Z\big)=E(Z^5)+12E(Z^4)+54E(Z^3)+108E(Z^2)+81E(Z) \]

Step 5: Standard normal moments.
\(E(Z)=E(Z^3)=E(Z^5)=0\); \(E(Z^2)=1\), \(E(Z^4)=3\).

Step 6: Substitute.
\[ 0+12(3)+0+108(1)+0=36+108=144 \]

Final Answer: \[ \boxed{144} \]